I think Hahn has to explore the FA market before making any trade. We know the biggest holes are C and 3B as well as set-up lefthander and some depth in bullpen. 1B may also be a hole if Paulie is gone. As fans, many think we need to upgrade SS, CF and to a lesser extent 2B. Of the biggest needs, maybe Semien plays 3B. So C is the biggest need at this point. If there is not a FA, then they need to trade for one.
But trading Alexei w/o an in-house replacement only opens another hole. Since the White Sox are pretty much a lousy team, trading may open up more holes than we have now unless Hahn can fleece someone. So free agency is the best method to fill holes or allow us to trade someone if we end up with a better option than we have. For example if you trade Reed and get a 2B, you can then trade Beckham. But now you have a closer opening.

I agree with you that trades are not a cure-all for a lousy team. But I don't know if any quality free agents will come to a team that lost 99 games. Players are money-driven but winning is still important to them. And I don't see the White Sox getting into a bidding war at the moment.